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A Levenberg-Marquardt method for large-scale bound-constrained nonlinear least-squares Shan, Shidong 2008

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A Levenberg-Marquardt Method For Large-Scale Bound-Constrained Nonlinear Least-Squares by Shidong Shan BSc (Hon.), Acad ia University, 2006 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Master of Science in The Faculty of Graduate Studies (Computer Science) The University of Br i t i sh Columbia (Vancouver) Ju ly 2008 © Shidong Shan 2008 Abstract T h e well known Levenberg-Marquardt method is used extensively for solving nonlinear least-squares problems. We describe an extension of the Levenberg- Marquardt method to problems wi th bound constraints on the variables. Each iteration of our algorithm approximately solves a linear least-squares problem subject to the original bound constraints. O u r approach is especially suited to large-scale problems whose functions are expensive to compute; only matr ix - vector products w i th the Jacobian are required. W e present the results of nu- merical experiments that illustrate the effectiveness of the approach. Moreover, we describe its application to a practical curve f i t t ing problem in fluorescence optical imaging. Contents Abstract i i Contents i i i List of Tables v List of Figures v i Acknowledgements v i i 1 Introduction 1 1.1 Overview 2 1.2 Definitions and notation 3 2 Background on Optimization 5 2.1 Unconstrained opt imizat ion 5 2.1.1 Line-search methods 6 2.1.2 Trust-region methods 6 2.2 Bound-constrained optimization 8 2.2.1 Active-set methods 8 2.2.2 Gradient projection methods 9 2.3 Algor i thms for bound-constrained optimization 10 3 Unconstrained Least Squares 12 3.1 Linear least squares 12 3.1.1 N o r m a l equations 12 3.1.2 T w o norm regularization 13 3.2 Nonlinear least squares 14 3.2.1 T h e Gauss-Newton method 14 3.2.2 The Levenberg-Marquardt method 16 3.2.3 U p d a t i n g the damping parameter 17 4 Nonlinear Least Squares W i t h Simple Bounds 20 4.1 Mot ivat ion 20 4.2 Framework 21 4.3 Stabi l izat ion strategy 24 4.4 T h e subproblem solver 25 4.5 The B C N L S algorithm 27 5 Numerical Experiments 30 5.1 Performance profile 30 5.2 Termination criteria 31 5.3 Benchmark tests w i t h A l g o r i t h m 566 31 5.4 C U T E r feasibility problems 34 6 Curve Fi t t ing in Fluorescence Optical Imaging 38 6.1 Brie f introduction to t ime domain optical imaging 38 6.2 Mathematica l models of fluorescence signals 40 6.3 Curve fitting 41 6.4 Fluorescence lifetime estimation 42 6.4.1 F i t t i n g parameters 42 6.4.2 Parameter bounds 43 6.4.3 Simulated data for lifetime fitting 43 6.4.4 Exper imental results 43 6.5 Fluorophore depth estimation 45 6.6 Opt ica l properties estimation 48 6.6.1 Simulation for estimating optical properties 49 6.6.2 Results for optical properties 50 6.7 Discussion 50 7 Conclusions and Future W o r k 52 Bibliography 53 List of Tables 5.1 Results of fifteen benchmark tests from A l g o r i t h m 566 . . . . . . 32 5.2 Summary of the benchmark tests from Algor i thm 566 33 5.3 Summary of C U T E r feasibility tests 36 6.1 Comparison of lifetime fitting w i t h and without D C component . 47 6.2 Comparison of lifetime fitting using B C N L S and A R T _ L M . . . . 48 6.3 Comparison of depth fitting results for (A) a l l data points, (B) data points w i t h depth > 1mm, (C) data points w i th S N R > 20, (D) data points w i th S N R > 20 and depth > 1mm 48 6.4 Comparison of depth fitting results for B C N L S and A R T X M for data points w i th S N R > 20 and depth > 1mm 49 6.5 and /x^ values used in s imulation 50 6.6 Comparison of results of optical properties w i th A R T _ L M and B C N L S 50 List of Figures 3.1 Geometrical interpretation of least squares 13 3.2 The image of function q{p) 18 5.1 Performance profiles on benchmark problems from A l g o r i t h m 566: number of function evaluations 34 5.2 Performance profiles on benchmark problems from Algor i thm 566: C P U time 35 5.3 Performance profile on C U T E r feasibility problems, number of function evaluations 37 5.4 Performance profile on C U T E r feasibility problems, C P U time . 37 6.1 T i m e domain optical imaging 39 6.2 Fluorescent decay curve 39 6.3 Fluorophore identification w i t h lifetime 40 6.4 A n example of fitted T P S F curve 44 6.5 Lifetime fitting results of simulated data w i th varying depth, without fitting D C component 45 6.6 Lifetime fitting results of simulated data w i th varying depth, w i th fitting D C component 46 6.7 Est imated depth versus true depth 47 6.8 Relative accuracy of calculated depth versus signal S N R 49 Acknowledgements I am grateful to my supervisor D r . Michael Friedlander, who made the subject of Numerica l Opt imizat i on exciting and beautiful. W i t h his knowledge, guidance, and assistance, I have managed to accomplish this work. I would like to thank D r . Ian Mi t che l l for his careful reading of this thesis and his helpful comments that have improved this thesis. I am fortunate to have had the financial support for this work from the N a t u r a l Sciences and Engineering Research Counc i l of Canada ( N S E R C ) i n the form of a postgraduate scholarship. Also , I a m thankful for the M I T A C S Accelerate B C internship opportunity at A R T Advanced Research Technologies Inc. It has been a privilege to work w i t h G u o b i n M a , Nicolas Robitai l le , and Simon Fortier i n the Research and Development group at A R T Advanced Re- search Technologies Inc. The i r work and ideas on curve f itting in fluorescence optical imaging have formed a strong foundation for Chapter 6 in this thesis. In addit ion, I wish to thank the three of them for reading a draft of Chapter 6 and their valuable comments and suggestions for improving the chapter. The Scientific C o m p u t i n g Laboratory at the University of B r i t i s h Co lumbia has been a pleasant place to work. I have enjoyed the support and company of my current and past colleagues: Ewout van den Berg , El izabeth Cross, Mar iso l Flores Garr ido , H u i Huang , D a n L i , Tracy L a u , and Jelena Sirovljevic. A m o n g them, Ewout van den Berg has always been of great help to me. I a m deeply indebted to my parents and family for their unconditional love and understanding. M y brother Hongqing has inspired me and set a great example for me to fulfi l l my dreams. Thanks are also due to many of my friends for their care and friendship over the years. Above a l l , I sincerely thank my best friend and partner B i l l Hue, whose love, support, and faithful prayers give me strength and courage to cope in difficult times. W i t h o u t h i m , I could have not achieved this far i n my academic endeavor. Final ly , I want to dedicate this thesis to my late father, who suddenly passed away while I was working on this thesis. In his life, my father was always passionate that his children would have every educational opportunity. There is no doubt that my father would be very proud i f he could live to witness another academic accomplishment of his youngest daughter. To my father, Kunsong Shan Chapter 1 Introduction T h i s thesis proposes an algorithm for nonlinear least-squares problems subject to simple bound constraints on the variables. T h e problem has the general form minimize 5l|r'(a;)||2 subject to (,<x<u, where i,u G E " are vectors of lower and upper bounds on x, and r is a vector of real-valued nonlinear functions " rx{x) we assume throughout that each ri{x) : E " —>• R is a twice continuously difFer- entiable nonlinear function. Nonlinear least-squares problems arise in many data-f itt ing applications in science and engineering. For instance, suppose that we model some process by a nonlinear function 0 that depends on some parameters x , and that we have the actual measurements j/j at time ti\ our a i m is to determine parameters x so that the discrepancy between the predicted and observed measurements is minimized. The discrepancy between the predicted and observed data can be expressed as r i (x ) = 4>{x,ti) - Vi. T h e n , the parameters x can be estimated by solving the nonlinear least-squares problem minimize |||T"(a;)||2- (1.1) Very large nonlinear least-squares problems arise in numerous areas of ap- plications, such as medical imaging, tomography, geophysics, and economics. In many instances, both the number of variables and the number of residuals are large. It is also very common that only the number of residuals is large. M a n y approaches exist for the solution of linear and nonlinear least-squares problems, especially for the case where the problems are unconstrained. See Bjorck [3] for a comprehensive discussion of algorithms for least squares, including detailed error analyses. Because of the high demand in the industry, nonhnear least-squares soft- ware is fairly prevalent. M a j o r numerical software libraries such as S A S , and programming environments such as Mathemat i ca and M a t l a b , contain nonlin- ear least-squares implementations. Other high quality implementations include, for example, D F N L P [50], M I N P A C K [38, 41], and N L 2 S 0 L [27]; see More and Wright [44, Chapter 3]. However, most research has focused on the nonlinear least-squares problems without constraints. For problems wi th constraints, the approaches are fewer (also see Bjorck [3]). In practice, the variables x often have some known lower and upper bounds from its physical or natural settings. T h e a im of this thesis is to develop an efficient solver for large-scale nonlinear least-squares problems wi th simple bounds on variables. 1.1 Overview T h e organization of the thesis is as follows. In the remainder of this chapter, we introduce some definitions and notation that we use in this thesis. Chapter 2 reviews some theoretical background on unconstrained and bound-constrained optimization. O n unconstrained opt imization, we summarize the basic strate- gies such as line-search and trust-region methods. For bound-constrained opt i - mizat ion, we review active-set and gradient-projection methods. We also briefly discuss some existing algorithms for solving the bound-constrained optimization problems, including A S T R A L [53], B C L S [18], and L - B F G S - B [57]. Chapter 3 focuses on unconstrained least-squares problems. We start w i th a brief review of linear least-squares problems and the normal equations. We also review two-norm regularization techniques for solving il l-conditioned linear systems. For nonlinear least-squares problems, we discuss two classical non- linear least squares algorithms: the Gauss-Newton method and the Levenberg- Marquardt method. We also explain the self-adaptive updating strategy for the damping parameter in the Levenberg-Marquardt method. These two methods and the updat ing strategy are closely related to the proposed algorithm in this thesis. Chapter 4 is the main contribution of this thesis. We explain our pro- posed algorithm, named B C N L S , for solving the bound-constrained nonlinear least-squares problems. We describe our algorithm wi th in the A S T R A L [53] framework. The B C N L S algorithm is based on solving a sequence of linear least-squares subproblems; the bounds i n the subproblem are copied verbatim from the original nonlinear problem. The B C L S least-squares solver forms the computational kernel of the approach. The results of numerical experiments are presented i n Chapter 5. In Chap - ter 6, we illustrate one real data-f itt ing application arising in fluorescence optical imaging. The B C N L S software package and its documentation are available at ht tp : / / w w w . cs. ubc. ca / labs / sc l /bcn ls / . 1.2 Definitions and notation In this section, we briefly define the notation used throughout the thesis. The gradient of the function f{x) is given by the n-vector g{x) = Vf{x) = and the Hessian is given by dfix)/dxx dfix)/dX2 df{x)/dx„ H{x) = Vy{x) = 9 ^ | V M 9 X 2 c Ilk. 120X1 dx'i I dx. H ^ X l dxidXn 0X2 oxn Sim T h e Jacobian of r{x) is the m-by-n matr ix J ( x ) = Vr{xf = Vr^ixf Vr2{xf For nonlinear least-squares problems, the objective function has the special form and, in this case, fix) = i r ( x ) ^ r ( x ) , g{x) = J{xfr{x), m H(x) = J{xfj{x) + '£^ri[x)V^ri{x). (1.2) (1.3) (1.4) Most specialized algorithms for nonlinear least-squares problems exploit the special structure of the nonlinear least-squares objective function (1.2). Our proposed algorithm also explores this structure and makes use of the above structural relations. We use the following abbreviations throughout: • Xk: the value of x at iteration fc; • x*: a local minimizer of / ; • ik,Uk- the lower and upper bounds for the subproblem at iteration k. • Tk = r{xk), vector values of r at x^; • fk = f{xk), function value of f &X Xk\ • Jk — J(xk), Jacobian of r{x.) at Xjt; • gk = g{xk), gradient of f{x) at x^; • Hk = H{xk), Hessian of / ( x ) at Xk- Chapter 2 Background on Optimization This chapter reviews some fundamental techniques for unconstrained and bound- constrained optimization. We begin wi th some basic strategies for unconstrained opt imizat ion problems, then we examine some specialized methods for bound- constrained problems. 2.1 UnconstrEiined optimization T h e general unconstrained optimization problem has the simple mathematical formulation minimize f{x). (2.1) x€R" Hereafter we assume that f{x) : R " —> E is a smooth real-valued function. B y Taylor 's theorem, a smooth function can be approximated by a quadratic model in some neighbourhood of each point x £ R " . The optimality conditions for unconstrained optimization problem (2.1) are well-known. We sunmiarize the conditions below without explanation. For more detailed descriptions, see Nocedal and Wright [47, Chapter 2]. Firs t -Order Necessary Conditions: If x ' is a local minimizer and / is continuously differentiable i n an open neighbourhood of x*, then g(x*) = 0. Second-Order Necesssiry Conditions: If x* is a local minimizer of / and V - ^ / exists and is continuous i n an open neighbourhood of x*, then ^(x*) = 0 and H{x*) is positive semidefinite. Second-Order Sufficient Conditions: Suppose that V ^ / is continuous in an open neighbourhood of x*, ^(x*) = 0, and H{x*) is positive definite. Then X* is a strict local minimizer of / . In general, algorithms for unconstrained optimization generate a sequence of iterates {xk}'^-Q using the rule Xk+\ = Xk+dk, where dk is a direction of descent. The two main algorithmic strategies are based on different subproblems for computing the updates dk- 2.1.1 Line-search methods Line search methods obtain a descent direction dk in two stages. F i rs t , dk is found such that gïdk < 0. (2.2) T h a t is, it requires that / is strictly decreasing along dk wi th in some neighbour- hood of Xfc. The most obvious choice for search direction is d^ = —g^, which is called the steepest-des cent direction. However, any direction dk that solves the equations Bkdk = —Qk for some positive definite matr ix Bk also satisfies (2.2). Second, the distance to move along dk is determined by a step length ak that approximately solves the one-dimensional minimizat ion problem minimize f(xk+adk)- a>0 Popular line search conditions require that ak gives sufficient decrease in / and ensures that the algorithm makes reasonable progress. For example, the Armijo condition imposes sufficient decrease by requiring that f{xk + akdk) < f{xk) + aiUkÇkdk, (2.3) for some constant cri € (0,1). The curvature condition imposes "reasonable progress" by requiring that ak satisfies gixk + akdkfdk > (r2gk<ik, (2.4) for (72 6 (o ' i , l ) - The sufficient decrease (2.3) and curvature conditions (2.4) collectively are known as the Wolfe conditions. T h e strong Wolfe conditions strengthen the curvature condition and require that , i n addit ion to (2.3), ak satisfies \g{xk + akdkfdk\ < cr2\gldk\. 2.1.2 Trust-region methods Trust-region methods were first developed for nonlinear least-squares problems [32, 37, 45]. W h e n f{x) is twice continuously differentiable, Taylor 's theorem gives f{x + d) = f + g'^d + \(fHd + 0{\\df)- Thus the quadratic model function ruk used at each iterate Xk is mk{d) = fk+ gld + \c[^Hkd. (2.5) The fundamental idea of trust-region methods is to define a trust-region radius for each subproblem. A t each iteration, we seek a solution dk of the subproblem based on the quadratic model (2.5) subject to some trusted region; minimize m^id) (2.6) subject to ||d||2 < A ^ . One of the key ingredients in a trust-region algorithm is the strategy for choosing the trust-region radius A ^ at each iteration. Given a step dk, we define the quantity ^ fjxk) - f(xk+dk) , - m f c ( 0 ) - m , ( 4 ) ' ^ ' which gives the ratio between the actual reduction and predicted reduction. T h e actual reduction represents the actual decrease of the objective function for the t r i a l step d^. The predicted reduction i n the denominator of (2.7) is the reduction predicted by the model function ruk- The choice of Afc is at least part ial ly determined by the ratio pk at previous iterations. Note that the predicted reduction should always be nonnegative. If pk is negative, the new objective value f{xi, + dk) IS greater than the current value fk, so the step must be rejected. If pk is close to 1, there is good agreement between the model and the function over the step, so it is safe to expand the trust region for the next iteration. If pk is positive but significantly smaller than 1, we do not alter the trust region, but if it is close to zero or negative, we shrink the trust region A ^ at next iteration. G iven some constants 771,772,71, and 72 that satisfy 0 < 771 < 772 < 1 and 0 < 71 < 72 < 1, we can update the trust-region radius by { (A/ i ,oo ) , if pk>m, [72Afc ,Afc], if pfc G [771,772], (2.8) ( 7 i A f c , 7 2 A f c ) , if Pk<m; See, e.g. Conn , G o u l d and Toint [9, Chapter 7]. Note that (2.6) can be solved equivalently as minimize Çkd + \(f{Hk + Xkl)d, for some positive Xk that is larger than the leftmost eigenvalue of Hk- Thus , the solution satisfies {Hk + \kl)d=-gk. (2.9) T h e relationship between (2.6) and (2.9) is used in Chapter 4 when we describe the B C N L S algorithm. For a summary of trust-region methods, see Nocedal a n d Wright [47, Chapter 4]. 2.2 Bound-constrained optimization B o u n d constrained optimization problems have the general form minimize f{x) (2.10) subject to £ < X < u, where l,u e R" are lower and upper bounds on the variables x. The feasible region is often called a "box" due to its rectangular shape. Note that both the lower and upper bounds may be optional, and when some components of x lack an upper or a lower bound, we set the appropriate components of ^ or w to — c » or +00, respectively. We define the set of b inding constraints [20, 53] by Bix*)={ i x*=ei, iîg{x*)i>0, or X* = Ui, if g{x*)i < 0, or k = Ui. Furthermore, the strictly binding set is defined as Bs(x*) = B{x*) n {i : g{x*)i ^ 0}. The standard first-order necessary condition for a local minimizer x* requires g{x')i = Q {0TiiB{x*). (2.11) T h e expression (2.11) shows the importance of the binding constraints to the bound-constrained opt imizat ion problems. T h e first-order necessary condition is also called the first-order Karush-Kuhn-Tucker ( K K T ) condition. T h e second- order sufficient condition for x* to be a local minimizer of the bound-constrained problem (2.10) is that it satisfies the K K T condition and also w^H{x*)w > 0, for a l l vectors w ^ 0 such that Wi — 0 for each i € Bs{x*). See [47, Chapter 12] for the detailed theory of constrained optimization. 2.2.1 Active-set methods Active-set methods for constrained optimization part i t ion the variables into active and inactive sets. Those variables whose indices are in the binding set are also classified as fixed variables, while the variables whose indices are not active are classified as free variables. Given any set of free variables, we can define the reduced gradient and the reduced Hessian matr ix , respectively, as the gradient and the Hessian matr ix of f{x) w i th respect to the free variables. In this terminology, the second-order sufficient condition requires that the reduced gradient be zero and that the reduced Hessian matr ix be positive definite at x*. In general, active-set algorithms for the solution of bound-constrained prob- lems use unconstrained minimizat ion techniques to explore the reduced problem defined by a set of free variables. Once this exploration is complete, a new set of free variables is chosen wi th the a im of dr iv ing the reduced gradient to zero. One drawback of the active-set method is that the working set changes slowly [47]. Usual ly at each iteration at most one constraint is added to or dropped from the working set. Thus the active-set method may require many iterations to converge on large-scale problems. For example, if there are ko ac- tive constraints at the starting point XQ, and kg constraints are active at the solution, then at least \ks — fco| iterations wi l l be required to reach the solution. 2.2.2 Gradient projection methods A gradient-projection method is a special type of active-set method that allows the active set to change rapidly from iteration to iteration. It is a very efficient method when there is an effective method for the projection onto the feasible set. For example, projection onto bound constraints can be done wi th linear cost. However, there are also other sets that can be efficiently projected onto, see, e.g.. Conn , Gou ld , and Toint [9, Chapter 12]. Gradient projection methods can be used to minimize both convex and non- convex functions; the feasible set, however, must be convex. One popular ap- proach to the gradient projection methods is to implement each iteration by a two-stage process. F i r s t , we compute an active face of the box by searching along the steepest descent direction —g from the current point x, and we com- pute a generalized Cauchy point by searching along the piecewise-linear path i n the feasible region. The working set is then defined to be the set of bound con- straints that are active at the Cauchy point. In the second stage, we compute a search direction using the active face by solving a subproblem i n which the active components of x are in the working set. Th is approach is implemented by B C L S [18], T R O N [33], Lancelot [8], L - B F G S - B [57], and some others. Given an arbitrary point x, the projection of x onto the feasible bounded region is defined as follows. The i t h component of the projection of x is given by r Ci if xi < £i, p{x,e,u)i = l Xi iîxi € [ei,ui], [ Ui if Xi > Ui. The projected gradient of the objective function / at a point x onto the feasible region is defined by gix) = P{x-g(x),ê,u)-x. (2.12) It is well-known that x* is a first-order stationary point for (2.10) if and only if gix*) = 0 [6], i.e., X* ^Pix* -g{x*),e,u). If the second-order sufficient condition can be satisfied, then the gradient- projection method is guaranteed to identify the active set at a solution in a finite number of iterations. After it has identified the correct active set, the gradient-projection algorithm reduces to the steepest-descent algorithm on the subspace of free variables. A s a result, this method is often used in conjunction w i t h a second-order method in order to achieve a faster rate of convergence. 2.3 Algorithms for bound-constrained optimization M a n y algorithms have been proposed to solve (2.10). For example, algorithms have been proposed by B y r d el al [5], C o n n et al [6, 7, 8], Hager and Zhang [23, 24], L i n and More [33], Zhu el al [57], and others. T h e main framework of recent approaches follows the two-stage process as described in Section 2.2.2. However, the algorithms differ on how to identify the active face and how to compute the search direction at each iteration. For example, the method in [6, 7] uses a two-norm trust-region method, and it computes the search step by applying the truncated conjugate-gradient method to the quadratic approximation of the objective function on the active face subject to the trust-region constraints. A similar approach is also used by L i n and More [33] in T R O N , but the t r u n - cated conjugate-gradient method is applied to the quadratic approximation of the objective function on the subspace parallel to the active face. The active-set method proposed by Hager and Zhang [23, 24] uses a backtracking line-search to identify the active face and apply the unconstrained method to the objective function on the active face. One well-known quasi-Newton method for solv- ing (2.10) is the L - B F G S - B algorithm [57], which computes a search direction by truncating the unconstrained L - B F G S [34] update relative to the subspace parallel to the active face. More recently, X u and Burke [53] have proposed the A S T R A L algorithm for solving large-scale nonlinear bound-constrained optimization problems. A S - T R A L is an active-set algorithm that uses both active-set identification tech- niques and l imited memory B F G S updat ing for the Hessian approximation. A t each iteration, A S T R A L uses a gradient projection step to determine an active face and then forms an ôo trust-region quadratic programming (QP) subprob- lem relative to the active face. The trust-region subproblems are solved using pr imal -dual interior point techniques. The A S T R A L algorithm has a close re- lationship to the proposed algorithm in this thesis, thus we wi l l describe the A S T R A L algorithm in more detail in Chapter 4. One particular software package for solving bound-constrained problems that plays an important role in this thesis is B C L S [18]. The B C L S algorithm solves the linear least-squares problem wi th simple bounds on the variables minimize ^\\Ax - b\\l + ^ô^\\x\\l + c'^x (2.13) subject to £ < X < u, where A is m x n (can be any shape), b G R"" , l,u,c e E " , and 5 is a damping parameter. In Chapter 4, we show that the problem (2.13) has the exact form for the subproblems to be solved in our proposed algorithm ( B C N L S ) for bound- constrained nonlinear least-squares. In fact, the B C N L S algorithm utilizes the B C L S package to solve the subproblem at each iteration. We explain the B C L S algorithm and describe how to use it effectively w i th in the B C N L S algorithm i n Chapter 4. Chapter 3 Unconstrained Least Squares T h i s chapter reviews some theoretical background on unconstrained least squares. We start wi th a preliminary review of linear least squares in Section 3.1, and we describe the two-norm regularization techniques for solving the i l l -conditioned linear systems. In Section 3.2, we describe two classical methods for uncon- strained nonlinear least squares problems: the Gauss-Newton method and the Levenberg-Marquardt method. We also describe a self-adaptive strategy for updat ing the damping parameters in the Levenberg-Marquardt method. 3.1 Linear least squares T h e classical linear least-squares problem has the form minimize I I I A x - f e U ^ , (3.1) where is an m x n matr ix , and b is an m-vector. Th i s problem belongs to a special class of opt imizat ion problems wi th much interest. First ly , the problem (3.1) is a special case of nonhnear least squares. Secondly, the classical methods for nonhnear least squares, such as the Gauss-Newton method and the Levenberg-Marquardt method in Section 3.2, iteratively solve hnear least- squares subproblems. Geometrically, to solve the problem (3.1), we want to find a vector x € R " such that the vector Ax € R™ is the closest point i n the range of A to the vector b. This geometrical interpretation is i l lustrated in Figure 3.1. The problem (3.1) has been studied extensively, and well-known numeri - cal algorithms exist. See Lawson and Hanson [29], Go lub and V a n Loan [21, Chapter 5], and Trefethen and B a u [52, Lecture 11] for more detail. 3.1.1 Normal equations For a given x, define the residual by r = b — Ax. A vector x s R " minimizes the norm of the residual if and only if r is orthogonal to the range of A: A^r = 0. Figure 3.1: Geometrical interpretation of least squares T h i s condition is satisfied if and only if À^M = A^b. (3.2) T h e equations in (3.2) is called the normal equations. Th i s system is nonsingular if and only if A has full rank. Consequently, the solution x is unique if and only if A has full rank. 3.1.2 Two norm regularization Regularization is a common technique for i l l -conditioned linear least squares problems [17]. The solutions of ill-posed problems may not be unique or may be sensitive to the problem data. The regularized least-squares problem has the quadratic form m i n i m K e ^\\Ax - bg + ^S'^\\x\\l (3.3) for some <5 > 0. Solving (3.3) is equivalent to solving the linear least squares problem 1 minuruze - x€R" 2 Note that the matr ix in (3.4) necessarily has full column rank. Historically, the study of problem (3.4) as a function of 5 has also been called ridge regression, damped least squares, or Tikhonov regularization [51]. The regularization is a technical device of changing the ill-posed problem to an approximate problem whose solutions may be well-behaved and preferable. For example, it may be preferable to have approximate solutions w i t h small norms. The intent of using 5 is to prevent ]|a;()2 from getting large when A is i l l -conditioned. B y performing singular value decomposition A = U'EV'^, we can show that Ml-tier/. A ' ' b âl X — 0 (3.4) where is the i t h column of the orthogonal matr ix U and CTJ is the i t h singular value of A. Note that increasing S would cause the norm \\xs || to decrease. Thus we can obtain some acceptable compromise between the size of the solution vector X and the size of the norm of the residuals by choosing proper 5. For further discussion about regularization and the choice of parameter S, refer to Lawson and Hanson [30, Chapter 25]. The two-norm regularization can also be regarded as a trust-region method. T h e solution x to equations (3.3) and (3.4) also solves mimmize i | | A E - 6 | | | (3.5) subject to ||xj|2 < A , for some A > 0. It can be shown that the solution of (3.4) is a solution of the trust-region problem (3.5) if and only i f x is feasible and there is a scalar 5 >0 such that x satisfies (3.4) and S{A - \\x\\2) = 0. See Nocedal and Wright [47, Chapter 10] for the proof. Instead of setting the trust region A directly, two-norm regularization mod- ifies A impl ic i t ly by adjusting the damping parameter S. The coimection be- tween the trust-region method and two-norm regularization is also revisited i n Section 3.2.2. Regularization techniques play a key role in this thesis. The B C L S algorithm in Section 4.4 solves a bound-constrained two-norm regularized linear least- squares problem. The Levenberg-Marquardt method discussed i n Section 3.2.2 solves a regularized linear least squares subproblem at each iteration. Moreover, the algorithm for the nonhnear least squares problems wi th simple bounds de- scribed in Chapter 4 solves a regularized linear least squares subproblem wi th bound constraints at each iteration. 3.2 Nonlinear least squares W i t h the theoretical background of opt imizat ion i n Chapter 2 and methods for linear least-squares problems from Section 3.1, we now describe two classi- cal methods for imconstrained nonlinear least-squares problems (1.1), namely, the Gauss-Newton method and the Levenberg-Marquardt method. For a more detailed description of these methods, refer to [47, Chapter 10]. 3.2.1 The Gauss-Newton method Perhaps one of the most important numerical methods is Newton's method. T h e Newton search direction is derived from the second-order Taylor series approximation to f{x + d), f(x + d)^f + g'^d+^cFHd. (3.6) Assume that H is positive definite. B y minimiz ing (3.6) at each step k, the Newton search direction is computed as the solution of the linear system Hd = -g. Newton's method requires H to be positive definite in order to guarantee that d is a descent direction. Also , H must be reasonably well-conditioned to ensiu-e that \\d\\ is small . W h e n H is positive definite, Newton's method typical ly has quadratic convergence. However, when H is singular, the Newton direction is not defined. Since Newton's method requhres computation of the second derivatives, it is only used when it is feasible to compute H. The Gauss-Newton method difi'ers from Newton's method by using an ap- proximation of the Hessian matr ix . For nonlinear least squares problems, i n - stead of using the Hessian i n (1.4), the Gauss-Newton method approximates the Hessian as H « J . (3.7) B y (1.3), we have g = J ^ r . Thus at each iteration, the Gauss-Newton method obtains a search direction by solving the equations j'^Jd^^-j'^r. (3.8) The Hessian approximation (3.7) of the Gauss-Newton method gives a num- ber of advantages over the Newton's method. F i r s t , the computation of J only involves the Jacobian computation, and it does not require any addit ional derivative evaluations of the indiv idual residual Hessians V^ri(a;) , which are needed in the second term of (1.4). Second, the residuals ri{x) tend to be small in many apphcations, thus the first term J in (1.4) dominates the second term, especially when Xk is close to the solution a;*. In this case, J is a close approximation to the Hessian and the convergence rate of Gauss-Newton is similar to that of Newton's method. T h i r d , J is always at least positive semi-definite. W h e n J has full rank and the gradient g is nonzero, the Gauss-Newton search direction is a descent direction and thus a suitable direction for a line search. Moreover, if r{x*) = 0 and Jix*)"^ J{x*) is positive definite, then x* is an isolated local min imum and the method is locally quadratically convergent. The fourth advantage of Gauss-Newton arises because the equations (3.8) are the normal equations for the linear least-squares problem minimize \\\Jd + r\\l. (3.9) d In principle, we can find the Gauss-Newton search direction by applying linear least-squares algorithms from Section 3.1 to the subproblem (3.9). The subproblem (3.9) also suggests another view of the Gauss-Newton method. If we consider a linear model for the vector function r (x ) as r{xk + d)xrk + Jkd, then the nonlinear least-squares problem can be approximated by l\\r(xk + d)g^l\\Jkd+r4l Thus, we obtain the Gauss-Newton search direction by minimiz ing the linear least-squares subproblem (3.9). Implementations of the Gauss-Newton method usually perform a line search in the search direction d, requiring the step length to satisfy conditions as those discussed i n Chapter 2, such as the A r m i j o and Wolfe conditions described in Section 2.1.1. W h e n the residuals are small and J has ful l rank, the Gauss- Newton method performs reasonably well. However, when J is rank-deficient or near rank-deficient, the Gauss-Newton method can experience numerical dif- ficulties. 3.2.2 The Levenberg-Mcirquardt method Levenberg [32] first proposed a damped Gauss-Newton method to avoid the weaknesses of the Gauss-Newton method when J is rank-deficient. Marquardt [37] extended Levenberg's idea by introducing a strategy for controlling the damping parameter. The Levenberg-Marquardt method is implemented in the software package M I N P A C K [38, 42]. The Levenberg-Marquardt method modifies the Gauss-Newton search direc- t ion by replacing J^J w i th J^J + S'^D ,where D is diagonal, 5 >0. Hereafter, we use D = I to simplify our description of the method. A t each iteration, the Levenberg-Marquardt search direction d satisfies iJ^J + Ô^I)d=-J^r. (3.10) The Levenberg-Marquardt method can also be described using the trust- region framework of Chapter 2. Let A be the two-norm of the solution to (3.10), then d is also the solution of the problem minimize |||</<̂  + ' "| l2 subject to \\d\\2 < A . Hence, the Levenberg-Marquardt method can be regarded as a trust-region method. The main difference between these two methods is that the trust-region method updates the trust region radius A directly, whereas the Levenberg- Marquardt method updates the parameter S, which in turn modifies the value of A implicit ly . For more description of the differences between the two methods, see [42, 43, 55, 56] for details. Note that the system (3.10) is the normal equations for the damped linear least-squares problem . . . 1 mimmize - d 2 J SI d + (3.11) The clamped least-squares problem (3.11) has the same form as (3.4) in Sec- t ion 3.1.2. Similar to the Gauss-Newton method, the Levenberg-Marquardt method solves the nonlinear least-squares problem by solving a sequence of l i n - ear least-squares subproblems (3.11). T h e local convergence of the Levenberg- Marquardt method is also similar to that of the Gauss-Newton method. The key strategy decision of the Levenberg-Marquardt method is how to choose and update the damping parameter S at each iteration. 3.2.3 Updating the damping parameter M u c h research has been done on how to update Ô. See, for example, [14, 36, 42, 46] for more details. From equations (3.10), if 5 is very small , the Levenberg- Marquardt step is similar to the Gauss-Newton step; and if (5 is very large, then d w —-^9, and the resulting step is very similar to the steepest descent direction. Thus , the choice of 6 i n the Levenberg-Marquardt method affects both the search direction and the length of the step d. If x is close to the solution, then we want the faster convergence of the Gauss-Newton method; if X is far from x", we would prefer the robustness of the steepest descent method. Similar to the trust-region method, the ratio pk (2.7) between the actual reduction of the objective function and the predicted reduction of the quadratic model is used to control and update the damping parameter 5k in the Levenberg- Marquardt method. In general, if the step dk is successful and pk is satisfactory, we accept dk and reduce S; otherwise, we reject dk and enlarge 5. Natmal ly , when Pk is close to 1 and the step dk is very successful, we may reduce Sk to near 0, then the next step would become an approximate Gauss-Newton step. O n the other hand, when dk is large and pk < 0, it is reasonable to enlarge Sk to move the search direction to a descent direction. We could adopt this strategy for updat ing the trust region radius A t and modify (2.8) for updating the damping parameter in the Levenberg-Marquardt method. For example, one possible approach, described by Fan [15], is T h e main drawback of this approach is that it is not sensitive to small changes in the values of pk- There exist jumps i n Sk-^-i/Sk across the thresholds of rji and 772 [15]. Recently, Yamashi ta and Pukushima [54] show that if the Levenberg-Marquardt parameter is chosen as and if the in i t ia l point is sufficiently close to the solution x*, then the Levenberg- Marquardt method converges quadratically to the solution x*. D a n et a l [10] propose an inexact Levenberg-Marquardt method for nonhnear least-squares problem wi th the parameter chosen as (5fc/4, if pk > Tj2, Sk, i f Pfc € [771,772], 4 4 , i f Pk<Vi- (3.12) Sk = I j r . l l , 2-6 \ \ 2 \ \ 1" 06 \ \ % 0.1 0.2 0.S 0.4 0.6 0,B 0-7 0.8 OJ Figure 3.2: The image of function q(p) where is a positive constant. T h e rationale of using a parameter u in adjusting the damping parameter is based on the following observations. W h e n the current iterate is too far away firom the solution and ||rfc|| is too large, ok could become too large; the resulting step dk could potentially be too small and prevent the sequence from converging quickly. O n the other hand, when the current position is sufficiently close to the solution, ||rfc|| maybe very small , so the 5k could have no effect. D a n et a l [10] show that the Levenberg-Marquardt method converges to the solution superlinearly when v G (0,2), and it converges quadratical ly when 1/ = 2. More recently, Fan and P a n [14] propose a strategy of adjusting the damping parameter by *I = afc||rfc||^ (3.13) for u € (0,2], and ak is some scaUng factor. To avoid the jumps in ôk+i/Sk across the thresholds of rji and T]2 and make more use of the information of the ratio pk, Fan and P a n use a scaling factor ak at each iteration, where ak+i is a function of pk and ak- Define the continuous nonnegative function g(p) = maoc{l/4,1 - 2(2p - 1)3}. (3.14) T h e image of function (3.14) is i l lustrated in Figure 3.2. The function of ak+i is then defined as ak+i = akq{pk)- (3.15) B y using the function q{p), the factor ak is updated at a variable rate ac- cording to the ratio pk, rather than s imply enlarging or reducing the Ô by some constant rate as in (3.12). Combining (3.13), (3.14) and (3.15) together, we have an updating strat- egy which is referred to as the self-adaptive Levenberg-Marquardt method [14]. T h i s technique is similar to the self-adaptive trust-region algorithm proposed by Hei [26]. Under the local error bound condition, Fan and P a n [14] show that the self-adaptive Levenberg-Marquardt method converges superlinearly to the solution for v € (0,1), while quadratical ly for v € [1, 2]. Note that the theoretical convergence proof of the self-adaptive updating strategy is for unconstrained nonlinear least-squares optimization. Nevertheless, in Chapter 4, we show that this updat ing strategy can be adapted and extended to bound-constrained nonlinear least-squares problems. Chapter 4 Nonlinear Least Squares With Simple Bounds This chapter presents the main contribution of this thesis. We describe our proposed algorithm ( B C N L S ) for solving the nonlinear least-squares problem wi th simple bounds minimize 5lk(a;)||2 (4.1) subject to i < X < u, where u € E " are vectors of lower and upper bounds on x, and r is a vector of real-valued nonlinear functions. In Chapter 2, we briefly introduced some algorithms for bound-constrained nonlinear optimization problems. In this chapter, we show how we adopt the general framework of the A S T R A L [53] algorithm into the B C N L S algorithm for solving the nonlinear least-squares problems. We also describe how we make use of the B C L S [18] algorithm to solve a sequence of bound-constrained hnear least-squares subproblems. We begin this chapter w i th a brief description of the motivation of our ap- proach. T h e n we illustrate the general framework of the proposed algorithm in Section 4.2. The strategies and techniques used for stabilizing and enhanc- ing the performance of the algorithm are described in Section 4.3. Section 4.4 describes the B C L S algorithm for solving the subproblems. F inal ly , in Sec- t i on 4.5, we show the detailed implementation steps of the B C N L S algorithm and summarize the basic features of the algorithm. 4.1 Motivation M a n y algorithms (for example, A S T R A L , L - B F G S - B ) exist for general nonlinear opt imizat ion w i th simple bounds, but such algorithms do not make use of any knowledge of the objective functions. O n the other hand, efficient methods (for instance, B C L S ) exist for solving hnear least-squares problems wi th simple bounds. Hence, we want to develop an efficient algorithm specially designed for large-scale nonlinear least-squares problems wi th simple bounds on variables. In the design of the B C N L S algorithm, we set the following three goals for our approach. F i r s t , we want an algorithm that scales well to large problems. Second, the algorithm should be very effective for the problems whose functions and gradients are expensive to evaluate. That is, the algorithm must be frugal w i th the evaluation of functions and gradients. T h i r d , we want to ensure that the variables are always feasible at each iteration, so that the intermediate results can be useful in practical applications. To achieve the above goals, we use the principles of the classical Gauss- Newton and Levenberg-Marquardt method for unconstrained nonlinear least- squares problem. Our algorithm ( B C N L S ) solves the bound-constrained nonhn- ear least-squares problem by iteratively solving a sequence of bound-constrained linear least-squares subproblems. T h e subproblems are handled by the software package B C L S . To fully exploit the advantages of B C L S , the subproblems are formed by using the two-norm regularization instead of trust-region method. A t each iteration, the damping parameter of the subproblem is updated using the self-adaptive updating strategy described in Section 3.2.3. 4.2 Framework T h e B C N L S algorithm uses the same quadratic model (3.6) for the objective function and the same Hessian approximation (3.7) as in the Gauss-Newton method. The algorithm ensures that the variables x ^— x + d are always feasible. T h a t is, we superimpose the simple bound constraints i<Xk + dk <u, (4.2) for each subproblem at step k. The bounds (4.2) can be written as ik<dk< Uk, (4.3) where £k = i — Xk, and Uk — u — Xk are the updated lower and upper bounds of d at iteration k. A s wi th the Gauss-Newton method, we take advantage of the special struc- ture (4.1) of the nonlinear least-squares objective. B y combining (3.9) and (4.3), we have extended the classical unconstrained Gauss-Newton method to the non- linear least-squares problem wi th simple bounds. The tentative subproblem has the form minimize \\\Jkdk + rk\\l (4.4) subject to £k < dk < Uk- However, if the subproblem (4.4) were to be solved without any modification, the proposed method may experience numerical difficulties when Jk is nearly rank-deficient. The numerical solution of (4.4) is dependent on the condition of J just as in the Gauss-Newton method. In Chapter 3, we have described how the trust-region method and the Levenberg-Marquardt method can be used to remedy the numerical difficul- ties of the Gauss-Newton method. It is natural for us to adopt these remedies. Recal l in Section 2.3, we have mentioned that trust-region methods have been used in several algorithms [6, 7, 53] for solving the nonlinear optimization prob- lems wi th simple bounds on variables. For bound-constrained problems, it is more convenient for the trust region to be defined by the || • ||oo norm, which gives rise to a box-constrained trust-region subproblem. For this reason, the ôo trust region is used in A S T R A L [53]. Our first approach is to mimic A S T R A L ' s approach by adding an ico trust- region constraint to the subproblem (4.4). minimize h\\Jkdk + rkWl (^-S) subject to £k < dk < Uk, \\d\\oo < Ak- The role of ||rf||oo < is to control the norm of d. We instead use the two-norm regularization to achieve the same effect. We can express the sub- problem (4.5) i n a two-norm regularized form minimize ||| J/tOÎ/fe + T"* : ! ! ! + l^fcNfcHi (4-6) subject to êk < dk < Uk, for some Sk > 0. This subproblem has the benefit that it can be solved directly by the software package B C L S [18]. One remaining challenge lies in how to adjust 5k at each iteration. For the algorithm to have a rapid convergence rate, it is important to have an effective strategy to update 5k at each step. Essentially, the algorithm uses 5k to control both the search direction and the step size. A lgor i thm 4.1 outlines the basic framework of our approach. The main com- putational component of this algorithm is solving the bound-constrained linear least-squares subproblem (4.6). In this way, the bounds are handled directly by the subproblem solver. Thus, we can apply the self-adaptive damping strat- egy described in Section 3.2.3, and the local and global convergence properties remain the same as in the unconstrained cases. A t this point, it is necessary to show that the proposed framework is a reasonable approach for solving the bound-constrained nonhnear least-squares problem (4.1). The key global and local convergence properties of A lgor i thm 4.1 can be established by showing that the B C N L S algorithm has a very shnUar framework as the A S T R A L algorithm. The A S T R A L algorithm solves a general bound-constrained nonlinear prob- lem (2.10) by solving a sequence of ôo trust-region subproblems. The subprob- lem has the basic template minimize ^S" Bkd + g^d (4.7) d subject to Ik < d < Uk, \\d\\oo < Ak. T h e objective function is based on the quadratic model (2.5), where Bk'isa sym- metric positive definite Hessian approximation of the objective function at step A l g o r i t h m 4.1: Outl ine of the B C N L S method 1 initialization: fc < — 0, a;o, 5o given. 2 while not optimal do 3 Solve the bound-constrained subproblem (4.6) for dk- 4 Compute the ratio pk (2.7). 5 if Pk is satisfactory then 6 Xk-+-i *- Xk + dk 7 else 8 1 Xk+l *— Xk- 9 end 10 Update the damping parameter Ok- 11 ie *-k+I. 12 end k. The A S T R A L algorithm solves the quadratic trust-region subproblem (4.7) by using a pr imal-dual interior point method. The outline of the A S T R A L algorithm is shown i n A lgor i thm 4.2. Note that the gradient projection restart procedure (line 6) m A lgor i thm 4.2 is a strategy to assist in the identification of the active constraints at the cur- rent solution. Under the standard nondegeneracy hypothesis, X u and Burke [53] established that the framework of A lgor i thm 4.2 converges both globally and locally to a first-order stationary point. Moreover, the sequence of {xk} gener- ated by the A S T R A L algorithm converges quadratically to x* under standard assumptions. The A S T R A L algorithm and its convergence theory follow the pattern estabhshed by Powell in [49] and has been used for similar algorithms from the Uterature [4, 6, 9, 47]. For detailed theoretical proof of the convergence properties, see [53]. After a l l , the nonlinear least squares problem (4.1) is a special case of the nonlinear problem (2.10). The similarity between the frameworks of Algo- r i t h m 4.1 and A l g o r i t h m 4.2 suggests that the convergence properties of the A S T R A L algorithm apply to the B C N L S algorithm for the bound-constrained nonlinear least-squares problem. The main differences lie in the specific tech- niques used for solving the quadratic subproblems. In this sense. A lgor i thm 4.1 specializes the A S T R A L algorithm to a special class of nonlinear least-squares opt imizat ion problems wi th simple bounds on variables. The new approach takes advantage of the special structure of the objective function by ut i l iz ing the special relationships between its gradient, Hessian and the Jacobian. Our approach eliminates the need to bui ld , compute and possibly modify the Hessian of the objective. Therefore, the B C N L S algorithm can make use of the function and gradient evaluations more efficiently than other similar approaches for the general nonlinear optimization problems. A l g o r i t h m 4.2: Outl ine of the A S T R A L method 1 initialization: Â; <— 0, l o given, SQ > 0, ÔQ min{maxi {u j — Zi},(5o}; 2 initial ize symmetric positive definite BQ € R . 3 choose a set of controlling constants. 4 while not optimal do 5 if ||dfc||oo is not acœptable then 6 Perform a gradient projection restart step. 7 end 8 Solve the trust-region subproblem (4.7) for dk- 9 Compute the ratio pk (2.7). 10 if Pk is satisfactory then 11  1 Xk+i ^ Xk + dk 12 else 13 Xk-^l Xk- 14 end 15 Update symmetric positive definite matr ix Bk- 16 Update the trust-region radius. 17 k*-k + l. 18 end 4.3 Stabilization strategy T h e A S T R A L algorithm makes use of the l^o trust-region on the search direction d for the subproblems. T h e £oo trust region constraint where e = ( 1 , 1 , . . . , 1)'^. B y superimposing the £oo trust region constraint onto the feasible bounds, the resulting feasible region continues to have a nice rectangular box shape, and the subproblems can be solved by applying the bound-constrained quadratic programming techniques. A t each iteration k, the A S T R A L algorithm updates the trust-region radius Ak based on the perfor- mance of the previous steps. The strategy of updating the trust-region radius prevents the norm of d from getting large and thus has the effect of stabilizing the algorithm. The B C N L S algorithm uses a very simUar strategy to the A S T R A L algo- r i t h m . Rather than updating the trust-radius directly, we use the two-norm reg- ularization technique and formulate the subproblem as the form of (4.6). The damping parameter 5k is then updated by using the self-adaptive Levenberg- Marquardt strategy in Section 3.2.3. The theoretical equivalence between the trust-region method and the two-norm regularization techniques has been es- tablished in Section 3.2.2 and Section 3.2.3. Compared w i th the trust-region l|4||oo <Afc is equivalent to approach, the two-norm regularization has the advantages of reducing the con- dit ion nunaber of the matr ix i n the subproblem and preventing from get- t ing large. Furthermore, the two-norm regularization strategy gives rise to a nicely formatted linear least-squares subproblem wi th simple bounds on the variables (4.6), which can be solved directly by B C L S . 4.4 The subproblem solver T h e kernel of the computational task in the B C N L S algorithm is the solution of the bound-constrained linear least-squares subproblems (4.6). The overall efficiency of the algorithm ult imately depends on the efficient solution of a se- quence of such subproblems. For large-scale problems, the Jacobian matrices are potentially expensive to compute and it would require relatively large storage space if we were to store the Jacobian matrices explicitly. To make the B C N L S algorithm amenable to large-scale nonlinear least-squares problems, we want to make sure that the optimization algorithms for the subproblem do not rely on matr ix factorizations and do not require the exphcit computation of Jacobian and the Hessian approximations. In the remainder of this section, we show that the software package B C L S has the desirable features to make it a viable choice for solving the subproblem in the B C N L S algorithm. B C L S is a separate implementation for solving linear least-squares problems w i t h bounded constraints. Descriptions of the software package B C L S can be found in [18], [19, Section 5.2], and [31, Section 2.4]. For the completeness of presenting the B C N L S algorithm, we reproduce a brief description of the B C L S algorithm here using the generic form of the problem (2.13). Because the hnear term c^x is not used i n the B C N L S algorithm, we set c = 0 and eliminate the linear term. Thus, the simplified version of the linear least-squares problem for the B C L S algorithm has the form mimmize ^\\Ax - bg + ^6^\\x\\l (4.8) subject to £ < X <u. The B C L S algorithm is based on a two-metric projection method (see, for example, [2, Chapter 2]). It can also be classified as an active-set method. A t al l iterations, the variables are partit ioned into two different sets: the set of free variables (denoted hy XB) and the set of fixed variables (denoted by xjv) , where the free and fixed variables are defined as i n classical active-set methods in Section 2.2.1. Correspondingly, the columns of matr ix A are also partit ioned into the free (denoted by ^ B ) and fixed (denoted by AAT) components based on the set of indices for the free and fixed variables. A t each iteration, the two-metric projection method generates independent descent directions AXB and AXN for the free and fixed variables. T h e search directions are generated from an approximate solution to the block-diagonal linear system AlA 0 -(52j 0 D AXB = Â^r - S^x, (4.9) where r = b — Ax is the current residual and D is a diagonal matr ix wi th strictly positive entries. The right-hand side of the equation (4.9) is the negative of the gradient of the objective in (4.8). T h e block-diagonal linear system (4.9) is equivalent to the following two separate systems {A%AB + S^I)AXB = AZr - S^XB DAXN = Aj-^r — 5 xjv. (4.10a) (4.10b) Thus, a Newton step (4.10a) is generated for the free variables XB, and a scaled steepest-descent step (4.106) is generated for the fixed variables x^- The ag- gregate step {AXB, AXN) is then projected onto the feasible region and the first minimizer is computed along the piecewise linear projected-search direc- tion. See [9] for a detailed description on projected search methods for bound- constrained problems. The linear system (4.9) is never formed explicit ly in the B C L S algorithm. Instead, AXB is computed equivalently as a solution to the least-squares problem minimize ^IABAXB - r\\l + hS'^llxs + A X B H ^ - Ax (4.11) The solution to (4.11) can be found by applying the conjugate-gradient type solver L S Q R [48] to the problem mmimize AXB As 01 AXB - (4.12) where (3 = max{(5, e}, and e is a smah positive constant. \i 5 < e, the re- sulting step is effectively a modified Newton step, and it has the advantage of safeguarding against rank-deficiency in the current sub-matrix AB- T h e major computational work i n the L S Q R algorithm is determined by the total number of matrix-vector products w i th the matr ix and its transpose. In order to con- tro l the amount of work carried out by L S Q R in solving the problem (4.12), the B C L S algorithm employs an inexact Newton strategy [11] to control the accuracy of the computed sub-problem solutions A X B - The following notable features make the B C L S algorithm a perfect fit for solving the subproblems (4.6) in the B C N L S algorithm. F irs t , the B C L S algo- r i t h m uses the matr ix A only as an operator. The user only needs to provide a routine to compute the matrix-vector products w i th matr ix A and its trans- pose. Thus the algorithm is amenable for solving linear least-squares problems in large scale. Second, the B C L S algorithm can take advantage of good starting points. The abi l i ty to warm-start a problem using an estimate of the solution is especially useful when a sequence of problems needs to be solved, and each is a small perturbation of the previous problem. This feature makes B C L S work well as a subproblem solver in the B C N L S algorithm. In general, the solutions to the subsequent subproblems are generally close to each other, and the solution of the previous iteration can serve as a good starting point for the current iter- ation and thus speed up the solution process. T h i r d , the bounded constraints are handled directly by the B C L S package, which leaves the B C N L S framework straightforward to understand and easy to implement. Fourth, the formulation of the two-norm regularized linear least-squares problem (4.8) enables the B C L S algorithm to fully exploit the good damping strategies. Therefore, the integra- t ion of the B C L S algorithm and the self-adaptive damping strategy contributes to the success of the B C N L S algorithm i n this thesis. 4.5 The B C N L S algorithm The implementation of the B C N L S algorithm requires the user to provide two user-defined function routines for a given bound-constrained nonlinear least- squares problem. The residual functions are defined by a user-suppUed routine called funObj, which evaluates the vector of residuals r(x) for a given x. The Jacobian of r{x) is defined by another user-written function called Jprod, whose essential function is to compute the matrix-vector products w i th Jacobian and its transpose, i.e., computes Jx and J^y for given vectors x and y w i th com- patible lengths for J . The in i t ia l point XQ and both lower and upper bounds are optional. If the in i t ia l starting point xo is provided but not feasible, then the infeasible starting point is first projected onto the feasible region and the projected feasible point is used as the in i t ia l point. The B C N L S algorithm wi l l output the solution x, the gradient g, and the projected gradient g of the objective function at x. In addition, it provides the user wi th the following information: the objective value at the solution, the total number of function evaluations, the number of iterations, the computing time, and the status of termination. A l g o r i t h m 4.3 describes our implementation of the B C N L S algorithm. A t the init ial izat ion stage, steps 1-5 define the damping updating strategj' for the given problem. The positive constant «tnin at step 5 is the lower bound of the damping parameter, it is used for numerical stabil ization and better computational results. It prevents the step from being too large when the sequence is near the solution. Step 7 sets the step acceptance criteria. Steps 8- 10 give the termination criteria for the algorithm. The algorithm is structured around two blocks. Steps 19-22 are executed only when the solution of the previous iteration is successful (i.e., when Xk has an updated value). Otherwise, the algorithm repeatedly solves the subproblem by steps 24-39 using an updated damping parameter Sf. unt i l pk is satisfactory. The B C N L S algorithm uses only one function evaluation and at most one gradient evaluation per major iteration. In this way, the algorithm ensures that the m i n i m u m number of function and gradient evaluations are used in the so- lut ion process. This approach may lead to more calls of B C L S for solving the A l g o r i t h m 4.3: Bound-Constrained Nonlinear Least Squares ( B C N L S ) input : funObj , Jprod , Xo, £, u output: x, g, g, info 1 Set damping parameter So > 0 2 Set scaling factor ao > 0 3 Set updating constant P Ç (0,2] 4 Define a nonnegative nonlinear function q{p) 5 Set the min imum scaling factor 0 < amin < cto 6 Set the starting point do for B C L S T Set step acceptance parameter 0 < po < 1 8 Set optimality tolerance € 9 Set max imum iterations 10 Set max imum function evaluations 11 A; +- 0 12 Xk +— XQ 13 dfc «— do 14 Compute the residuals <— funObj (xk) 15 Sk aollrfcll". 16 Set iStepAccepted < — true 17 while ll̂ fcll > e do 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 38 36 37 38 39 40 e n d 41 x*-Xk, g*-9k, 9^9k- if iStepAccepted then Compute the gradient gk <— J p r o d ( r t , 2) Compute the projected gradient gk <— P{xk — gk,i) u) — Xk i k - i - x k Uk*-U-Xk end [dk] *- BCLSiJprod, - r^ , 4 , Uk, do, 0, Sk) Compute new residuals rnew *— funObj (x^ + dk) Compute the ratio pk (2.7). if pk > Po then Xk+i *- Xk+dk rk+l *— ''new do *— dk IStepAccepted else Xk+l <- Xk Tk+l Tk IStepAccepted end Set ak+\ max{Q;min,a!fcg(pfc)}. Update 5k+i ^ ak+iWrkW"• k^k + 1. true false. subproblems and potentially use more computation of the matrix-vector prod- ucts w i th Jacobian and its transpose. However, in general, large-scale problems requires more computing time to evaluate the function and gradient values. Therefore, the B C N L S algorithm can perform well for large-scale problems. Our implementation of A lgor i thm 4.3 shows that we have achieved our p r i - mary goals for the bound-constrained nonlinear least-squares solver. The B C - N L S algorithm has the following basic features. F i r s t , two-norm regularization and damping techniques are used in solving the subproblems. Second, Hessian approximations are never formed and no second-order derivatives are required to be computed. T h i r d , the simple bounds are handled directly by the sub- problem solver B C L S and the sequence of {xk} is always feasible at any time. Fourth , the B C L S algorithm makes use of the conjugate-gradient type method L S Q R . B y inherit ing the significant features from the B C L S and L S Q R soft- ware packages, the B C N L S algorithm uses the Jacobian only as an operator. It only requires the user to provide matrix-vector products for the Jacobian and its transpose. Thus the B C N L S package is amenable for bound-constrained nonlinear least-squares problems i n large-scale. Chapter 5 Numerical Experiments We give the results of numerical experiments on two sets of nonlinear least- squares problems. The first set is composed of the fifteen benchmark nonlinear least-squares problems from the Net l ib collection A lgor i thm 566 [1, 39, 40]. The second set of test problems are generated from the C U T E r testing environment [22]. Us ing these two sets of test problems, we compare the performances of B C - N L S , L - B F G S - B and A S T R A L . The L - B F G S - B Fortran subroutines were down- loaded from [5, 57]. The A S T R A L M a t l a b source codes were downloaded from [53]. The comparisons are based on the number of function evaluations and C P U times. 5.1 Performance profile To il lustrate the performance, we use the performance profile technique de- scribed i n Dolan and More [12]. The benchmark results are generated by run- ning the three solvers on a set of problems and recording information of interest such as the number of function evaluations and C P U time. Assume that we are interested in using C P U time as a performance measure, we define tp.^ as the C P U time required to solve problem p by solver s. Let ibest be the min imum C P U time used by a l l solvers for problem p. The performance ratio is given by r - Is^ tbest T h e ideas are the same when comparisons are based on the number of function evaluations. The performance profile for solver s is the percentage of problems that a performance ratio rp^g is w i th in a factor T € R of the best possible ratio. A plot of the performance profile reveals a l l of the major performance characteristics. In particular, if the set of problems is suitably large, then solvers wi th large probabil ity psir) are to be preferred. One important property of performance profiles is that they are insensitive to the results on a small number of problems. A lso , they are largely unaffected by small changes in results over many problems [12]. In this chapter, al l the performance profile figures are plotted in the log j - scale. The M a t l a b performance profile script perf.m was downloaded from http : / /www-unix .mcs .anl .gov /~more /cops [12]. If a solver s fails on a given problem p, its corresponding measures, such as the number of function evalu- ations or C P U time, are set to N a N . In this way, the performance profile not only captures the performance characteristics, but also it shows the percentage of solved problems. 5.2 Termination criteria In our numerical experiments, each algorithm is started at the same in i t ia l point. We use the following criteria to terminate the algorithms: < emach-10^ < e < e (*) < < 1000, where emach is the machine epsUon, e is the opt imal i ty tolerance, gk is the projected gradient (2.12) of the objective function, n / is the number of function evaluations. Note that the termination criteria are the same as in the A S T R A L and L - B F G S - B solvers, except that (*) is partictilar to the B C N L S solver. If an algorithm terminates but the optimality condition < e is not satisfied, we consider the algorithm to have failed on the given problem. However, since the objective f imction is in the form of least-squares, i f an algorithm terminates w i t h /fc < e and Xk is feasible, we consider the algorithm succeeded on the given problem, regardless of the value of the projected gradient. 5.3 Benchmarlc tests with Algorithm 566 T h e test set of benchmark least-squares problems from Nethb A lgor i thm 566 are unconstrained least-squares problems. We do not include Problems 1-3 from the original Nethb collection, as these are hnear least-squares problems. The remaining fifteen nonlinear least-squares problems are used as the benchmark to test the three boimd-constrained algorithms. We arbitrari ly choose the bounds on the variables to be 0 < i < oo. The original Net l ib problems were implemented in Fortran subroutines. We recoded the problems in M a t lab. Furthermore, we use the standard in i t ia l points as defined i n A l g o r i t h m 566 for a l l the test problems. The results w i th fifteen test problems are given in Table 5.1. To summarize and compare the overall performances of the three solvers based on the results in Table 5.1, we use the following criteria to judge the outcomes (success or failure) of the solvers on any problem. If both / > 10~^ and > 10""*, then we consider the solver has failed on the problem, otherwise it is a success. Based on the criteria, we show the overall performance of the three algorithms in Table 5.2. \\fk-fk-i\\ max(l,||Mi,||A_i||) Il5.ll fk m rif Table 5.1: Results of fifteen benchmark tests from Algor i thm 566 Problem Algor i thm 9\\ rif time(Sec) B C N L S L B F G S B A S T R A L 2.47e - 15 5.08e - 06 7.42e - 06 1.30e - 07 3 . 5 1 e - 0 1 5.36e - 02 5 3.00e - 02 22 1.12e - 02 36 5.20e - 01 B C N L S L B F G S B A S T R A L 3.62e + 02 3.62e + 02 5.31e + 01 2.50e + 02 2.50e + 02 1.29e - 03 1 9.70e - 01 22 1.43e - 02 22 3.59e - 01 B C N L S L B F G S B A S T R A L 9.98e - 30 6.82e - 06 9.83e - 06 9.99e - 15 1.05e - 02 1.27e - 03 7 4.00e - 02 21 1.51e - 02 27 5.79e - 01 B C N L S L B F G S B A S T R A L 6.40e + 01 6.40e + 01 6.40e + 01 4.62e - 07 2 . 1 3 e - 1 4 1.74e - 06 4 2.00e - 02 5 7.95e - 03 3 5.53e - 02 B C N L S L B F G S B A S T R A L 4 . 1 1 e - 0 3 4 . 1 1 e - 0 3 4.11e - 03 1.28e - 05 3.46e - 06 3.52e - 06 6 4.00e 37 2.07e 86 1.16e - 0 2 - 0 2 + 00 B C N L S 4 L B F G S B A S T R A L 1 . 5 4 e - 0 4 1.54e - 04 1.54e - 04 3.29e - 06 1.08e - 06 6.90e - 06 9 7.00e 34 2.18e 72 6.33e - 0 2 - 0 2 - 0 1 10 B C N L S L B F G S B A S T R A L 1.21e + 04 7.30e + 08 5.03e + 06 2.69e + 05 4.06e + 10 2.86e + 07 21 1.40e 4 8.36e 31 5.55e - 0 1 - 0 3 - 0 1 11 B C N L S L B F G S B A S T R A L 6.88e - 03 6.88e - 03 6.88e - 03 1.53e - 06 6.61e - 06 2.72e - 05 6 7.00e 33 2.95e 45 6.36e - 02 - 0 2 - 0 1 12 B C N L S L B F G S B A S T R A L 5.65e - 08 5 . 1 7 e - 0 6 9.30e - 06 2.66e - 04 8.43e - 03 3.08e - 03 5 4.00e 21 1.48e 32 5.54e - 0 2 - 0 2 - 0 1 13 B C N L S L B F G S B A S T R A L 6.22e + 01 6.22e + 01 6.22e + 01 3.52e - 03 1.60e - 05 1.14e - 02 18 l.OOe 21 1.31e 37 5.21e - 0 1 - 0 2 - 0 1 14 B C N L S L B F G S B A S T R A L 1.06e 1.06e 1.06e •05 -05 •05 1.14e + 00 1.57e - 03 1.66e - 03 15 9.00e 30 2.10e 17 3.61e - 0 2 - 0 2 - 0 1 15 150 B C N L S L B F G S B A S T R A L 7.39e - 06 4.81e - 02 4.40e - 02 2.33e - 04 1.87e + 00 2.29e + 00 86 l .O le 145 2.29e 201 5.42e + 02 + 00 + 00 B C N L S 1 .22e - 14 2.50e - 07 3 5.31e + 00 16 2000 L B F G S B l.OOe + 09 2.00e + 06 4 4.83e + 02 A S T R A L l , 4 3 e - 22 6.90e - 1 0 14 1.64e + 01 Continued on next page Table 5.1 - continued from previous page Problem n A lgor i thm / \\9 I  time(Sec) B C N L S 2.54e - 02 2.57e - 0 3 11 l.OOe - 01 17 5 L B F G S B 2.54e - 02 2.18e - 0 6 40 3.07e - 01 A S T R A L 2.54e - 02 2.52e - 0 2 76 1.37e + 00 B C N L S 2 . 0 1 e - 02 8.45e - 0 6 15 2.30e - 01 18 11 L B F G S B 2.01e - 02 4.90e - 0 6 73 3.36e - 01 A S T R A L 2.01e - 02 5.35e - 0 5 357 5.65e + 00 Solver Success Failure Success Rate(%) B C N L S 10 5 67 L B F G S B 10 5 67 A S T R A L 9 6 60 Table 5.2: Summary of the benchmark tests from Algor i thm 566 Overal l , the three algorithms have similar success rates on this set of test problems. For Problem 15 {Chebyquad function), both L - B F G S - B and A S - T R A L seem to have difficulties in finding the solution, but B C N L S has per- formed reasonably well on the problem, terminated wi th / = 7.39 x 10"^ and (l̂ ll = 2.33 X 10" ' ' . A lso , a l l three solvers failed on Problem ^{Helical Valley Function), Prob lem 10 {the Meyer function) and Problem 14 (the Brown and Dennis function). For Problem 16 [the Brown almost-linear function), both B C N L S and A S T R A L performed well, but I ^ B F G S - B d id not solve i t . O n the other hand, for Prob lem 17 {the Osborne 1 function), L - B F G S - B performed bet- ter than A S T R A L and B C N L S . For details on these function definitions, refer to [1, 39, 40]. Because both B C N L S and A S T R A L were written in M a t l a b , whereas L - B F G S - B was written in Fortran, we compare the number of function eval- uations among the three algorithms. T h e computing time is compared only between B C N L S and A S T R A L . Figure 5.1 and Figure 5.2 illustrate the perfor- mance profiles for B C N L S , L - B F G S - B , and A S T R A L on the fifteen benchmark nonhnear least-squares problems. Figure 5.1 compares the number of function evaluations among the three solvers. We observe that the performance profile for B C N L S hes above both L - B F G S - B and A S T R A L for T < 2.6. T h a t is, for approximately 55% of the problems, the performance ratio of the function evaluation for the B C N L S algorithm is wi th in a factor 2.6 of the best possible ratio. Therefore, the performance profile of the number of function evaluations for B C N L S dominates that of L - B F G S - B and A S T R A L . We conclude that B C - N L S is more efficient in terms of function evaluation. Figure 5.2 compares C P U time between B C N L S and A S T R A L . Clearly, the performance profile of C P U time for B C N L S dominates A S T R A L , thus B C N L S uses less C P U time than A S T R A L . Performance profiles, number of function evaluations 0 .4 ; 0.À -A — • — B C N L S - • - L B F G S B • - D - A S T R A L 9 ^ <>•• I «& t 9" - 6 o-è - 6 •—-a 1 ,H - i 9 2 . 5 3 X Figure 5.1: Performance profiles on benchmark problems from A l g o r i t h m 566: number of function evaluations. 5.4 C U T E r feasibility problems T h e general constrained C U T E r test problems have the form minimize f(x) , (5.1) subject to 4 < c(x) <Uc, (•x <x <Ux, where a; G M " , c{x) is a vector of m nonlinear real-valued functions of R " —>• R™, 4 and Uc are lower and upper bounds on c{x), and ix and Ux are lower and upper bounds on x. We use problems from the C U T E r set to generate a set of nonhnear least- squares test problems. These problems are cast as that of finding a feasible point to (5.1). Such problems can arise as subproblems for more general opt imizat ion packages, where it is important to have a feasible starting point. We can rewrite the constrained problem (5.1) as minimize fix) (5.2) subject to c{x) — s = 0, (•X ^ X < Ux- Because we are only interested in getting a feasible point for the C U T E r prob- Performance profiles, CPU time 0) o.e( • Oi SS S 0 . 9 • 0 .3 0 .2 0.1' 0 • -6 • -<b — B C N L S - a - A S T R A L ^ - 6 • -6 I 9 -o - 6 2 .6 3 Figure 5.2: Performance profiles on benchmark problems from Algor i thm 566: C P U time. lem, we can s imply ignore the objective function i n (5.2), and deal only w i th the constraints i n (5.2). The feasibility problem can then be expressed as mmimize subject to è l | c ( a . ) - * l l i ic<S< Uc, To fit this problem statement into the standard bound-constrained nonhnear least-squares framework, we define X = u = Ux Ur c{x) = c{x) — S. Now, the feasibility problem has the standard form mmimize subject to él|c(5)iii, i<x<û. (5.3) Clearly, the solution to (5.3) satisfies al l the constraints in the original C U T E r problem (5.1). We refer to the problems generated from C U T E r in such a way as the C U T E r feasibility problems. To select the testing problems from the C U T E r environment, we have used the C U T E r selection ut i l i ty [16]. The basic criterion for the problem selection Solver Success Failure Success rate(%) B C N L S 183 14 92.9 L B F G S B 151 46 76.7 A S T R A L 166 31 84.3 Table 5.3; Summary of C U T E r feasibility tests is that the C U T E r problem must have at least one nonlinear constraint. We do not impose any addit ional restriction on the objective function, the number of variables, bound conditions, and so on. We installed both the C U T E r and SifDec environment in large scale on a L i n u x Pent ium 4 P C wi th 3 G H z processor and I G B R A M . SifDec can compile a l l the selected problems. However, inside the M a t l a b C U T E r testing environment, the call to csetup would fail for large problems due to an "out of memory" error. We thus exclude very large test problems from the selected test list. A s a result, a total of 197 C U T E r feasibility problems were tested on the three solvers B C N L S , L - B F G S - B , and A S T R A L . The max imum number of variables i n the tested problems is 700, and the maximum number of constraints c{x) is 284. To compare the performances of the three solvers, we use the same criteria as in Section 5.3 to judge the success and failure of a solver on any problem. T h e overall outcomes for the solvers is given in Table 5.3. For this set of test problems, B C N L S has the highest success rate compared wi th L - B F G S - B and A S T R A L . It has solved more problems than either L - B F G S - B or A S T R A L . The outcomes in both Table 5.2 and Table 5.3 suggest that the overall success rate of B C N L S tends to increase as the size of the test set of nonlinear least-squares problems increases. Figure 5.3 illustrates the performance profiles for the number of function evaluations for B C N L S , L - B F G S - B , and A S T R A L . We can see from Figure 5.3 that the curve of B C N L S lies above the curves of both L - B F G S - B and A S T R A L . T h e performance profile shows that B C N L S uses fewer function evaluations than both L - B F G S - B and A S T R A L in solving bound-constrained nonlinear least- squares problems. Figure 5.4 shows the performance profile for C P U time for B C N L S and A S T R A L . Because the performance profile of B C N L S dominates that of A S T R A L , Figure 5.4 shows that B C N L S is more efficient than A S T R A L in C P U time. For large-scale problems, the cost of function evaluations usually dominates the computing time. The numerical experiments in the two sets of test problems show that B C N L S outperforms its counterparts in solving bound-constrained nonlinear least-squares problems. A s the size of the test set and the number of variables increase, the benefits of B C N L S become more promising. The results are not surprising to us, because B C N L S takes advantage of the special structure of the nonlinear least-squares functions and thus is efficient in its use of function evaluations, whereas both L - B F G S - B and A S T R A L do not use any structure information about the functions. Performance profile, number of function evaluations X Figure 5.3: Performance profile on C U T E r feasibility problems, number of func- t ion evaluations Performance profile, CPU time X Figure 5.4: Performance profile on C U T E r feasibility problems, C P U time Chapter 6 Curve Fitting in Fluorescence Optical Imaging T h i s chapter describes one industr ial application of the B C N L S solver. In col- laboration w i th A R T Advanced Research Technologies Inc., a company that designs molecular imaging products for the medical and pharmaceutical indus- tries, we have applied B C N L S for solving the nonlinear inverse problems i n t ime-domain (TD) fluorescence optical- imaging processes. It is a common practice in research and industry to estimate parameters through a curve fitting procedure. B y comparing the measured signal w i th some model data, one can estimate a combination of parameters that minimizes the difference between measured data and the model signals through the general curve fitting techniques, such as least squares. In this chapter we describe three curve fitting applications that use B C N L S : estimation of fluorescence lifetime, fluorophore depth, and optical properties of a fluorophore. Our description of the T D fluorescence problem is based on [13]. 6.1 Brief introduction to time domain optical imaging In smal l -animal fluorescence imaging, fluorophores or labelled biomolecules are injected into small animals such as mice. W h e n an external light source i l l u m i - nates a region of interest, the fluorophore transitions into an excited state and then re-emits a low-energy photon upon relaxation. The photon can then be detected by some optical detectors to generate fluorescence images. T D optical imaging uses photon-counting technology. It measures the arrival t ime of each detected photon. Figure 6.1 illustrates a typical configuration of a T D optical imaging system [28]. T h e fluorescent decay curve is fundamental to the T D fluorescent optical imaging techniques. Figure 6.2 illustrates a typical fluorescent decay curve, which is also commonly called the temporal point-spread function ( T P S F ) [28]. T h e T P S F curve contains important information about the fluorophore: • depth: the location of the fluorophore; Figure 6.1: T i m e domain optical imaging loboo t i ine (ns) Figure 6.2: Fluorescent decay curve • concentration: the fluorophore concentration; • fluorescence lifetime: the t ime for the intensity to decay by a factor of e. Fluorescence lifetime is also defined as the average t ime a fluorophore re- mains in its excited state following excitation. Fluorescent lifetime is an in t r in - sic property of the fluorophore that depends on the nature of the fluorescent molecule and its environment. Fluorophore identification is therefore possible using fluorescence lifetime. T h e relationship between the hfetime and the slope of the T P S F is shown i n Figure 6.3. In general, the shorter the lifetime of a fluorophore, the steeper the falling slope of the T P S F curve. The t ime-domain approach provides a unique opportunity to distinguish fluorescent signals w i t h identical spectra based on the differences i n their fluorescent lifetime. W i t h T D technology, fluorophore depth can be estimated from the temporal character- istics of the T P S F , and fluorophore concentration can be calculated from the intensity and photon depth. In addit ion, T D technology allows 3D reconstruc- 1 2 3 4 s 6 7 8 8 10 11 12 Figure 6.3: Fluorophore identification w i t h lifetime t ion of fluorophore distributions. 6.2 Mathematical models of fluorescence signals In fluorescence lifetime imaging, lifetime is measured at each pixel and displayed as contrast. In t ime domain, the fluorescence signal Fo{t) is a sum of several decay curves over time: F o ( t ) = ^ A i e - t ( 6 . 1 ) i where t represents t ime. Ai and T , are the amplitude and lifetime of the ith fluorophore component, and DC represents the offset signal in the data. The amplitude is related to many characteristics of a fluorophore, such as quan- t u m yie ld , extinction coefficient, concentration, volume, excitation and emission spectra, and so forth. In addit ion, the combination of the temporal profile of the excitation laser pulse and the system impulse response function ( I R F ) , denoted by S{t), also contributes to the signal. Thus , the measured signal F{t) can be modeled as F{t) = Fo{t)*S{t), where * denotes the convolution operator. A convolution is an integral that expresses the amount of overlap of one func- t i on g as it is shifted over another function / . Mathematical ly , the convolution of two function / and g over a finite range [0, t] is given by [f*9m= f f{r)g{t-T)dT. Jo W h e n fluorophore is embedded inside a bulk tissue or turb id medium, there w i l l be two more terms contributing to the convolution: the propagation of excitation light from source to fluorophore, denoted by Hifg — ff,t), and the propagation of fluorescent light from fluorophore to detector, denoted by E{ff - f d , i ) ; see [13]. Let f represent a 3-dimensional location vector. The locations of fluorophore, light source, and detector are represented by f / , fg, and rv, respectively. The fluorescence signal F{t) inside tissue has the form F(t) = H{fs - ff, t) * Fo{t) * E(ff - fd, t) * S{t); see [35]. B o t h the H{rs — rf,t) and E{ff — fd,t) are very complicated functions of many parameters, which require knowledge of tissue optical properties, such as the absorption coefficient Ha and the reduced scattering coefficient p'^, etc, and the spatial information of the fluorophore. Such information is usually not available in practical applications. For a detailed description of the equations, refer to [25, 35]. However, for a fluorescence signal from small-volume biological tissues—such as those obtained in mice—the light diffusion due to photon prop- agation, H{fs — ff,t) and E{ff —rd,t), does not significantly change the shape of the temporal profile of a fluorescence signal, while the photon propagation affects the amplitude and the peak position of fluorescence decay curve. Under the condition of a relatively short optical path between the excitation and fluorescent signal, such as a fluorophore inside a mouse, the effect of light diffusion in tissues can be simplified as a time delay ùd. That is, F{t)^ FQ{t)*5{M)*S{t). (6.2) In practice, a measured signal usually contains a D C component. B y substi- tut ing (6.1) into (6.2), we can express the mathematical model for fluorescence signal as F{t)KDC + 5{At)*S(t)*YlAie-^i. (6.3) i Hereafter i n this chapter, al l numerical experiments are based on the mathe- matical model (6.3) for the curve fitting applications. 6.3 Curve fitting In general, curve fitting is the process of minimiz ing the difi'erence between the observed data yd[ti) and the data from mathematical model ym{ti) over some unknown parameters. Let ri = j-iyd{ti) - VmiU)) be the residual between the observed data and predicted model, where cXi is a weighting term. The objective used for curve fitting is then X^ = \\\r{x)\\l = Y.i [J-(2/<i(i^)-1/™(t.)) In practice, the unknown parameters usually have known lower and upper bounds from their physical settings or based on a priori knowledge of the prob- lem. Thus, the bound-constrained curve fitting problem i n T D fluorescence optical imaging has the standard general form of (4.1). For a l l experiments on simulated data, we follow A R T ' s research approach to estimate the fluorescence lifetime, depth, and optical properties separately through curve f itt ing procedure based on the mathematical model in Section 6.2. It would be interesting to combine a l l three separate curve fitting problems into one and find an optimized combination of a l l parameters. The optimization problem wi l l remain the same in theory wi th the only difference of increased number of unknown parameters. In Section 6.4, 6.5, and 6.6, we describe our experiments of applying the B C - N L S solver to estimate the fluorescence lifetime, depth, and optical properties in T D technology. T h e results are presented in comparisons wi th A R T ' s imple- mentation of Levenberg-Marquardt algorithm ( A R T _ L M ) . To precisely compare two optimization algorithms, both methods must use the identical stopping c r i - teria. A s a bound-constrained opt imizat ion solver, B C N L S uses an optimality condition based on the projected gradient of the objective function. Conversely, A R T J L M is an unconstrained optimization method which has an optimality condition based on the gradient of the objective function. In order to compare the computational efficiency, we need to adjust the stopping criteria in different algorithms so that two algorithms have the same termination conditions. This change may slightly affect the reported numerical results in this chapter. 6.4 Fluorescence lifetime estimation In small -animal fluorescence lifetime imaging, measurements of fluorescence life- t ime can yield information on the molecular microenvironment of a fluorescent molecule. M a n y factors in the microenvironment have effect on the lifetime of a fluorophore. T h e measurements of lifetime can be used as indicators of those factors. In in vivo studies, these factors can provide valuable diagnostic infor- mation relating to the functional status of diseases. Furthermore, the lifetime measurements represent considerable practical advantages, as the measurements are generally absolute, being independent of the fluorophore concentrations. 6.4.1 Fitting parameters In lifetime fitting procedure, the parameters that need to be estimated include Ai, Ti, At, and DC. For a multiple lifetime fluorescence signal composed of n fluorophores, there w i l l be 2n -I- 2 parameters. However, one can use two strategies to reduce the number of fitting parameters by two. One strategy is to estimate the D C component separately; another is to use relative amplitude ai inste.ad of absolute amplitude A^. The normalized amplitude ai is given by This allows one of the relative amplitudes to be determined by the normal- ization feature of Q J , i.e., Yli'^i ~ 1- C)ur numerical experiments adopt both of these strategies, and so the total number of f itting parameters is 2n for a fluorescence signal F{t) composed of n fluorophore components. 6.4.2 Parauneter bounds In fluorescence lifetime fitting, the unknown parameters have known bounds: Tmin ^ fi ^ Tmux 0 < a i < 1 0<At< r / 6 0 < D C < Cnaax /3 , where T is the size of the time window of a T P S F , Cmax is the count maximum of a T P S F , Train = 0.25ns, rmax = T/i. The bounds of and the lower bound of D C originate from their physical l imits . A l l other bounds are practical l imits in order to increase the reUabihty of fitting results. 6.4.3 Simulated data for lifetime fitting A U data used in the experiments are generated and provided by A R T . Several sets of simulated data were used to estimate parameters through solving the nonlinear least squares optimization problem (4.1). Each data set was generated by varying one of the following parameter: • fluorophore depth d • signal D C level • relative amplitude a , • lifetime gap A T • mean lifetime f • maximal signal amplitude Cmax', the other parameters are held fixed. In order to obtain statistics of fitting results, random Poisson noise was generated and added to the fluorescence signal, and multiple trials were performed for each set of simulation. 6.4.4 Experimental results Three sets of error measurements are defined to characterize the performance of different optimization methods for curve fitting. The figure of merit includes the fitting error, standard deviation, and error margin of fitted values. The fitting error reflects the fitting accuracy. T h e standard deviation indicates the T=1.00 1.84ns, f=0.26 0.74, DC=-2, t0=0.16 ns time (ns) Figure 6.4: A n example of fitted T P S F curve repeatability of the fitting. E r r o r margin refers to the max imum error i n each data set. T h e error margin measures the robustness of the fitting method. Figure 6.4 shows one example of the curve fitted by B C N L S i n fluorescence lifetime fitting process. In this example, the known parameters in the simulated data are set as n = 0.98, T 2 = 1 .82 , / i = 0.25, /2 = 0.75, DC = 0, and the computed parameters are n = 1,T2 = 1 .84 , / i = 0.26, /2 = 0.74, DC = —2, and the final objective value = 1-0003. For a data set w i t h about 1000 data points, the fitted errors of reconstructed values are w i th in reasonable l imits for an industry problem. Some typical results for fifetime fitting are i l lustrated i n Figure 6.5 and F i g - ure 6.6, where Figure 6.5 results from not treating D C as a fitting parameter, and Figure 6.6 results from including D C as a fitting parameter. In both cases, the same simulated data file w i t h varying fluorophore depth was used. In F i g - ures 6.5 and 6.6, the computed results for each fltting parameter are presented i n terms of the error of fitted values related to the true values (errTr), the error margin (errMax) , the standard deviation (std), and the objective value at the solution (x^). Table 6.1 summarizes the effect of D C component on fluorophore lifetime fitting results from Figures 6.5 and 6.6. It shows that estimating D C component separately has the effect of reducing overall fitting errors and hence improve the accuracy of the fitting results. T h i s suggests that reducing the number of fitting parameters results i n fewer degrees of freedom of the system and increases the l ikelihood of the fitting parameters being close to the true value. For numerical examples of A R T ' s implementation of Levenberg-Marquardt 1.4 1.2 1 0 . 8 1 0.8 r i 0 . 6 0 . 4 0 . 2 I 1 0 . 5 0 - 0 . 5 - 1 e r r T r = 0 . 0 9 1 , eiTMax=0.51, s t d = 0 . 1 8 e r r T r = 0 . 0 9 2 , e r r M a x = 0 . 3 8 , s t d = 0 . 1 2 s t d B a r t rue 6 1 0 d e p t h e r r T r = 0 . 0 9 7 , e r r M a x = 0 . 3 8 , s t d = 0 . 1 5 11 S t d B a r t rue 5 1 0 d e p t t i e r r T r = 0 . 0 0 0 , e r r lV lax=0 .00 , s t d = 0 . 0 0 X S t d B a r • t rue 5 10 d e p t h 5 10 d e p t h e r r T r = 0 . 0 9 7 , e r rMax= !0 .38 . s t d = 0 . 1 5 I I r T X S t d B a r t rue 11 1 1 j 1 t ; Figure 6.5: Lifet ime f itt ing results of simulated data w i th varying depth, without f i t t ing D C component algorithms ( A R T i M ) , refer to [13]. Table 6.2 shows a comparison of B C N L S and A R T _ L M on the same simulated data. Note that B C N L S gets better results in terms of fitting errors. However, A R T X M takes slightly less computation t ime compared w i t h B C N L S . Th i s is l ikely due to the parameter transformation used i n A R T X M to handle the bound constraints indirectly. 6.5 Fluorophore depth estimation Fluorescent inclusion depth can be estimated by fiuorescent T P S F peak position if the hfetime of the fluorophore and medium optical properties are known a priori [25]. Due to noise, it is not accurate to simply take the peak position from the T P S F . Instead, a curve fitting procedure is employed to determine the depth. To avoid the instabil i ty due to noise at the beginning and ending parts of the T P S F , where the count level is low and relative noise level is high, we estimate fluorophore depth by only using the data port ion near the peak of the T P S F . T h e depth estimation also permits recovery of the fluorophore concentra- t ion . The fitting parameters i n the fluorophore depth estimation consist of fluorophore depth, and amplitude. T h e bounds of a l l the parameters come from their physical l imitations. Simi lar to the lifetime fitting procedure, simulated data w i th random noise were generated and added to the fiuorescence signal. A curve fitting procedure e r r T i = 0 . 1 2 7 , e r r M a x = 0 . 4 6 , s td=0 .11 1.3 ^ 1-2 I 1.1 1 0 . 9 1 s - 0 . 8 1 0 . 6 0 .4 ( 0 - 3 e r r T i ^ O . 2 2 6 , e r r M a x = 1 . 1 5 , s td=0 .22 s t d B a r t rue 5 1 0 16 d e p t h e r r T r = 0 . 1 5 8 , e r r M a x = 0 . 4 2 , s t d = 0 . 1 3 S t d B a r t rue 5 10 16 d e p t h e r r T r = 1 . 7 4 8 , e r r M a x = 5 . 3 5 , s t d = 0 . 9 6 S t d B a r t rue 5 1 0 16 d e p t h S t d B a r t rue 5 1 0 1 5 d e p t h e r r T R O . 1 6 8 , e r r M a x = 0 . 4 2 , s t d = 0 . 1 3 X S t d B a r • t rue • 11 Figure 6.6: Lifet ime fitting results of simulated data w i t h varying depth, w i t h fitting D C component is used to attempt to recover the known fluorophore depth and concentration. Similarly , error margin and standard deviation are used to compare the depth fitting results for different algorithms. Table 6.3 shows the depth fitting re- sults using B C N L S for one set of benchmark data. T h e abbreviations used i n Table 6.3 are • d: the calculated depth; • con: the computed fluorophore concentration; • m n E r r M g n : the average of the error margin over a l l data sets; • r M n M g n ( % ) : the average of the ratio of error margin to true value over a l l statistic data sets; • r M a x M g n ( % ) : the m a x i m u m of the ratio of error margin to true value over a l l statistic data sets; • meanStd: the average of the standard deviation over a l l data sets; • rMeanStd (%) : the average of the ratio of std to true value over a l l data sets; • r M a x S t d ( % ) : the m a x i m u m of the ratio of std to true value over a l l data sets. ^- Chapter 6. Curve Fitting in Fluorescence Optical Imaging 47 s- Parameter Comparison N o D C W i t h D C errTr 0.09 0.13 n e r r M a x 0.51 0.46 std 0.18 0.11 errTr 0.09 0.23 T2 e r r M a x 0.38 1.15 std 0.12 0.22 errTr 0.10 0.16 ai e r r M a x 0.38 0.42 std 0.15 0.13 errTr 0.00 1.75 DC e r r M a x 0.00 5.35 std 0.00 0.96 Table 6.1: Comparison of lifetime f itt ing w i t h and without D C component er rCen=0 .02 , e r rMax=0 .11 , std=0.06 * 1 o e r ro rMaxbar X S tdBar • true 1 1 "2 4 6 8 10 12 tn jBDept i l Figure 6.7: Est imated depth versus true depth In Table 6.3, we compare the depth fitting results for four sets of data points: (A) a l l data points, (B) data points w i t h depth greater than 1mm, (C) data points w i t h S N R greater than 20, (D) data points w i th S N R greater than 20 and depth greater than 1mm. The results show that better results can be obtained i n the depth fitting procedure by selecting those data points w i th S N R > 20 and depth > 1mm. Table 6.4 compares the depth estimation results from B C N L S and A R T _ L M for the latter case. Notice that a l l error margins and standard deviations obtained w i th B C N L S are very close to those obtained wi th the A R T X M algorithm. Figure 6.7 illustrates the estimated depth compared w i th true depth. F i g - ure 6.8 shows the relationship between the error of the estimated depth and the signal to noise ratio S N R . In general, signal w i th higher signal to noise ratio results i n more accurate estimated depth. These results by B C N L S also match w i t h the results from A R T _ L M very closely. Parameter Comparison B C N L S A R T X M errTr 0.07 0.07 . n errMax 0.17 0.19 std 0.09 0.11 errTr 0.13 0.19 T2 errMax 0.35 0.52 std 0.17 0.23 errTr 0.11 0.11 Oil errMax 0.29 0.28 std 0.14 0.15 errTr 1.53 2.92 DC e r r M a x 2.59 7.71 std 0.83 2.96 C P U time (seconds) 43.00 36.00 Table 6.2: Comparison of lifetime fitting using B C N L S and A R T X M D a t a points A l l (A) d > 1mm (B) S N R > 20 (C) (B) & (C) d con a con d con d con m n E r r M g n 0.28 64.56 0.21 63.04 0.17 34.75 0.11 34.91 r M n M g n 25.41 24.47 3.84 23.88 19.75 13.15 1.96 13.24 r M a x M g n 259.45 216.93 30.89 90.73 178.82 44.91 8.63 44.91 meanStd 0.16 21.54 0.12 21.59 0.11 13.48 0.06 13.71 r M e a n S t d 15.61 8.33 2.22 8.30 12.56 5.12 1.17 5.20 r M a x S t d 139.94 92.98 15.23 49.86 93.62 15.19 5.29 15.19 Table 6.3: Comparison of depth fitting results for (A) a l l data points, (B) data points w i th depth > 1mm, (C) data points with S N R > 20, (D) data points w i t h S N R > 20 and depth > 1mm 6.6 Optical properties estimation T h e effective optical properties, absorption coefficient /Xa and reduced scattering coefficient / i ^ , of biological tissues play a crit ical role in most algorithms for processing optical fluorescence data. For small animals such as mice, there can be large variations of the optical properties wi th in a given region of interest. In estimation of optical properties, the curve f itt ing problem has the same general form as the curve fitting problem in Section 6.3. The unknown param- eters are the optical properties /i» and /x^, and the amplitude A. The lower and upper bounds of the optical properties are based on a priori knowledge of the medium optical properties values. In our experiments, these bounds are set as 0.0001 <p.a< 0.2 0.2 <p'^< 3.5 0.85 <A< 1.15, „ . Depth Concentration compar ison ^ç^^^g A R T X M B C N L S A R T X M m n E r r M g n 0.11 0.11 34.91 34.91 r M n M g n 1.96 2.00 13.24 13.20 r M a x M g n 8.63 8.60 44.91 44.90 meanStd 0.06 0.06 13.71 13.71 r M e a n S t d 1.17 1.20 5.20 5.20 r M a x S t d 5.29 5.30 15.19 15.20 Table 6.4: Comparison of depth f itt ing results for B C N L S and A R T _ L M for data points w i t h S N R > 20 and depth > 1mm 11r - 10 r E r r C e n = 0 . 5 % , rE r rMax=2 .0%, rS td=1 .2% ^ • 0 e r r o r M a x b a r i X S tdBar true 2 5 30 3 5 4 0 4 5 5 0 b e s t S N R Figure 6.8: Relative accuracy of calculated depth versus signal S N R where the units of M o and p'^ are mm~^. 6.6.1 Simulation for estimating optical properties A n I R F curve w i th a peak count of 1000 and a ful l -width-at-half -max ( F W H M , or the time between 50% of the m a x i m u m count level before and after the peak) of 400 picoseconds was numerically generated. The peak count of the T P S F s was fixed to be 1500. Different T P S F curves were calculated using each pair of {Ha, p's) combination from the pre-selected sets of values i n Table 6.5. For comparisons, one set of tests was carried out without noise. For each pair of (/Xa, Ms), multiple trials were performed by adding Poisson noise (shot noise) to the T P S F and I R F curves. The mean values of the fitted optical properties and standard deviation for a l l the trials were then computed. The relative errors are the differences between the fitted values and their corresponding true values are shown in Table 6.5. Parameter Values (mm )̂ Ma 0.001, 0.005, 0.01, 0.03, 0.05, 0.1 0.2, 0.5, 1, 2, 3 Table 6.5: fia and p'^ values used in simulation without noise w i th noise Comparison Statistic A R T _ L M B C N L S A R T _ L M B C N L S Residual M e a n S T D 0.00 0.00 0.00 0.00 65.60 1.95 65.71 1.96 T i m e (s) M e a n 0.15 0.17 0.13 0.16 S T D 0.03 0.08 0.13 0.03 R e l Pa (%) M e a n -0.00 0.00 -1.48 -0.22 S T D 0.00 0.00 4.56 4.10 Re l (%) M e a n S T D 0.00 0.00 0.00 0.00 -0.35 0.79 0.00 0.75 Table 6.6: Comparison of results of optical properties wi th A R T i M and B C - N L S 6.6.2 Results for optical properties We have compared the performance of the solver B C N L S wi th A R T X M , note that no parameter transform was used wi th A R T _ L M in this case. The compar- ison criteria for different algorithms include the relative error, residual function value at the solution, and computation time. Table 6.6 shows the results of optical properties for the simulated data. For the data without Poisson noise, B C N L S performs as well as A R T X M method. However, for the simulations w i th Poisson noise, B C N L S outperforms the A R T X M method. 6.7 Discussion T h e numerical experiments on estimation of fluorescence lifetime, depth and optical properties at A R T illustrate that B C N L S can be applied to solve the curve fitting problems w i th bounds on parameters in practice. The challenges i n fluorescence lifetime fitting lie i n multiple lifetime fitting. A l though the ex- perimental results presented in this chapter are l imited to fluorescent signals of two fluorophore components, B C N L S can be easily adapted to the cases for multiple fluorescence lifetime curve fitting. Moreover, curve fitting is a process of finding an optimized combination of unknown parameters by some optimization method. In theory, there are some differences between the solutions of curve fitting opt imizat ion and the recon- struction of the known values i n the simulation. G iven the mathematical model for fluorescence signal in Section 6.2, our experiments reveal that a good curve fitting optimization solution may not always result in satisfactory reconstruction of the unknown. Nevertheless, the experiments show that the B C N L S solver is able to handle the simple bounds on variables in this curve fitting application. Chapter 7 Conclusions and Future Work In this thesis, we have developed the B C N L S algorithm for bound-constrained nonhnear least squares that solves a sequence of bound-constrained linear least squares subproblems, using techniques motivated by the Levenberg-Marquardt method and ^2-norm regularization. The subproblems are solved by B C L S , an existing solver for bound-constrained hnear least squares. Our approach differs substantially from previous general bound-constrained optimization methods. Our approach takes advantage of the special structure of the objective and uses the two-norm regularization to formulate the subproblems. B C N L S efficiently uses of function evaluations at the expense of potentially solving more linear linear-squares subproblems. The convergence properties of the new method are generally as good as, or better than, quasi-Newton methods such as L - B F G S - B , or the trust-region approach such as A S T R A L . Prehminary results are promising. The industr ial apphcation of curve f itting in fluorescence optical imaging fleld suggests that the new algorithm may prove to be useful i n practice. There is much scope for further development of the algorithm described i n this thesis. One promising direction would be to allow the user to choose preconditioners to speed up the process of solving the subproblems for large-scale systems. 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